For a small scale, one relatively secure, but somewhat inflexible, approach is to generate a large batch (say 1 million) public keys (use a deterministic wallet like Electrum uses so they're all governed by one secret key) and just store them on the payment platform site, issuing one per order. You have a distinct address and no private keys have to be online.

If you want the whole deal of getting USD or similar in your bank automatically, I would suggest investigating payment processor services that wrap all of that for you.